Is the `Iddah (waiting period) of the divorced effective from the time the divorce took place, or from the time of registering it at the court?
                                It is considered effective from the very moment her divorce took place, but not from the time of registering it at the court. And Allah Knows Best.
                             
                        
                            
                                
                                Should Al-Basmalah (i.e. saying Bismillaaahi Ar-Rahmaani Ar-Raheemi) be recited after Takbeerat Al-Ihram (saying Allahuakbar) which marks the beginning of prayer?
                                All perfect praise be to Allah,The Lord of The Worlds                                                                                                                                                                        A praying person is obliged to recite Al-Basmalah at the beginning of Chapter Al-Faatihah, and in every Rak`ah (unit of prayer) because it is a verse of it. And Allah Knows Best.
                             
                        
                            
                                
                                I joined a savings club and was scheduled to receive the payout first. My friend asked to take my place in line, so I requested 50 dinars in exchange for letting her have my turn. What is the ruling on this?
 
 
 
 
 
                                This is impermissible, as it involves taking money without providing something of value in return according to Islamic law. And Allah The Almighty Knows Best.
                             
                        
                            
                                
                                After I had prayed, I found a drop of menstrual blood and was unsure when that happened; what is the ruling on my prayer and fast?
                                All praise is due to Allah, peace and blessing be upon the Messenger of Allah.
Your prayer and fast are correct ,and you should not make up for them, and we ask Allah, The Almighty, to duely reward you for these acts of worship. Your menses start once you see blood because the situation remains as such unless there is certainty or predominant conjecture, as per the Fiqh maxim: "Certainty is not removed by doubt". And Allah Knows Best.